Partial flag varieties and preprojective algebras

[Variétés de drapeaux partiels et algèbres préprojectives]

Geiß, Christof ; Leclerc, Bernard ; Schröer, Jan

Annales de l'Institut Fourier, Tome 58 (2008), p. 825-876 / Harvested from Numdam

Résumé

Soit $\Lambda $ une algèbre préprojective de type $A, D, E$, et soit $G$ le groupe algébrique complexe semi-simple et simplement connexe correspondant. Nous étudions les modules rigides des sous-catégories ${\rm Sub\,} Q$ où $Q$ désigne un $\Lambda $-module injectif, et nous introduisons une opération de mutation sur les modules rigides complets de ${\rm Sub\,} Q$. Ceci conduit à des structures d’algèbre amassée sur les anneaux de coordonnées des variétés de drapeaux partiels associées à $G$.

Abstract

Let $\Lambda $ be a preprojective algebra of type $A, D, E$, and let $G$ be the corresponding semisimple simply connected complex algebraic group. We study rigid modules in subcategories ${\rm Sub\,} Q$ for $Q$ an injective $\Lambda $-module, and we introduce a mutation operation between complete rigid modules in ${\rm Sub\,} Q$. This yields cluster algebra structures on the coordinate rings of the partial flag varieties attached to $G$.

Détail

Publié le : 2008-01-01

MR 2427512 | Zbl 1151.16009 | 1 citation dans Numdam

DOI : https://doi.org/10.5802/aif.2371
Classification:  14M15,  16D90,  16G20,  16G70,  17B10,  20G05,  20G20,  20G42
Mots clés: variété de drapeaux, algèbre préprojective, catégorie de Frobenius, module rigide, mutation, algèbre amassée, base semi-canonique

Bibliographie

[1] Auslander, M.; Smalø, S. Almost split sequences in subcategories, J. Algebra, Tome 69 (1981), pp. 426-454 | Article | MR 617088 | Zbl 0457.16017

[2] Auslander, Maurice; Reiten, Idun Homologically finite subcategories, Representations of algebras and related topics (Kyoto, 1990), Cambridge Univ. Press (London Math. Soc. Lecture Note Ser.) Tome 168 (1992), pp. 1-42 | MR 1211476 | Zbl 0774.16005

[3] Bautista, R.; Martinez, R. Representations of partially ordered sets and 1-Gorenstein Artin algebras, Proceedings, Conference on Ring Theory, Antwerp, 1978, Dekker (1979), pp. 385-433 | MR 563305 | Zbl 0707.16003

[4] Berenstein, A.; Fomin, S.; Zelevinsky, A. Cluster algebras III. Upper bounds and double Bruhat cells, Duke Math. J., Tome 126 (2005), pp. 1-52 | Article | MR 2110627 | Zbl pre02147024

[5] Borel, A. Linear algebraic groups, Springer, 2nd Enlarged Edition (1991) | MR 1102012 | Zbl 0726.20030

[6] Bourbaki, N. Groupes et algèbres de Lie, Hermann, chap 4, 5, 6 (1968) | MR 240238 | Zbl 0483.22001

[7] Buan, A.; Iyama, O.; Reiten, I.; Scott, J. Cluster structures for 2-Calabi-Yau categories and unipotent groups (arXiv:math.RT/0701557)

[8] Buan, A.; Marsh, R.; Reineke, M.; Reiten, I.; Todorov, G. Tilting theory and cluster combinatorics, Adv. Math., Tome 204 (2006), pp. 572-618 | Article | MR 2249625 | Zbl 1127.16011

[9] Fomin, S.; Zelevinsky, A. Double Bruhat cells and total positivity, J. Amer. Math. Soc., Tome 12 (1999), pp. 335-380 | Article | MR 1652878 | Zbl 0913.22011

[10] Fomin, S.; Zelevinsky, A. Cluster algebras. I. Foundations, J. Amer. Math. Soc., Tome 15 (2002), pp. 497-529 | Article | MR 1887642 | Zbl 1021.16017

[11] Fomin, S.; Zelevinsky, A. Cluster algebras II. Finite type classification, Invent. Math., Tome 154 (2003), pp. 63-121 | Article | MR 2004457 | Zbl 1054.17024

[12] Geiß, C.; Leclerc, B.; Schröer, J. Cluster algebra structures and semicanonical bases for unipotent subgroups (arXiv:math.RT/0703039)

[13] Geiß, C.; Leclerc, B.; Schröer, J. Semicanonical bases and preprojective algebras, Ann. Sci. Ecole Norm. Sup., Tome 38 (2005), pp. 193-253 | Numdam | MR 2144987

[14] Geiß, C.; Leclerc, B.; Schröer, J. Rigid modules over preprojective algebras, Invent. Math., Tome 165 (2006), pp. 589-632 | Article | MR 2242628 | Zbl pre05057631

[15] Geiß, C.; Leclerc, B.; Schröer, J. Verma modules and preprojective algebras, Nagoya Math. J., Tome 182 (2006), pp. 241-258 | MR 2235343 | Zbl 1137.17021

[16] Geiß, C.; Leclerc, B.; Schröer, J. Auslander algebras and initial seeds for cluster algebras, J. London Math. Soc., Tome 75 (2007), pp. 718-740 | Article | MR 2352732 | Zbl pre05196120

[17] Geiß, C.; Leclerc, B.; Schröer, J. Semicanonical bases and preprojective algebras II: A multiplication formula, Compositio Math., Tome 143 (2007), pp. 1313-1334 | Article | MR 2360317 | Zbl pre05206856

[18] Geiß, C.; Schröer, J. Extension-orthogonal components of preprojective varieties, Trans. Amer. Math. Soc., Tome 357 (2005), pp. 1953-1962 | Article | MR 2115084 | Zbl 1111.14045

[19] Gekhtman, M.; Shapiro, M.; Vainshtein, A. Cluster algebras and Poisson geometry, Moscow Math. J., Tome 3 (2003), pp. 899-934 | MR 2078567 | Zbl 1057.53064

[20] Happel, D. Triangulated categories in the representation theory of finite-dimensional algebras, Cambridge University Press, Cambridge, London Mathematical Society Lecture Note Series 119 (1988) (x+208pp) | MR 935124 | Zbl 0635.16017

[21] Hoshino, M. On splitting torsion theories induced by tilting modules, Comm. Alg., Tome 11 (1983), pp. 427-439 | Article | MR 689417 | Zbl 0506.16018

[22] Kashiwara, M.; Saito, Y. Geometric construction of crystal bases, Duke Math. J., Tome 89 (1997), pp. 9-36 | Article | MR 1458969 | Zbl 0901.17006

[23] Keller, B. On triangulated orbit categories, Doc. Math., Tome 10 (2005), pp. 551-581 ((electronic)) | MR 2184464 | Zbl 1086.18006

[24] Kleiner, M. Approximations and almost split sequences in homologically finite subcategories, J. Algebra, Tome 198 (1997), pp. 135-163 | Article | MR 1482979 | Zbl 0903.16014

[25] Lakshmibai, V.; Gonciulea, N. Flag varieties, Hermann, Travaux en cours, Tome 63 (2001) | Zbl 1136.14306

[26] Lusztig, G. Canonical bases arising from quantized enveloping algebras, J. Amer. Math. Soc., Tome 3 (1990), pp. 447-498 | Article | MR 1035415 | Zbl 0703.17008

[27] Lusztig, G. Quivers, perverse sheaves, and quantized enveloping algebras, J. Amer. Math. Soc., Tome 4 (1991), pp. 365-421 | Article | MR 1088333 | Zbl 0738.17011

[28] Lusztig, G. Introduction to quantum groups, Birkhäuser (1993) | MR 1227098 | Zbl 0788.17010

[29] Postnikov, A. Total positivity, grassmannians and networks (arXiv:math.CO/0609764.)

[30] Ringel, M. C. The preprojective algebra of a quiver, Algebras and Modules II, (Geiranger, 1996), CMS Conf. Proc., Tome 24 (1998), pp. 467-480 (AMS) | MR 1648647 | Zbl 0928.16012

[31] Scott, J. Grassmannians and cluster algebras, Proc. London Math. Soc., Tome 92 (2006), pp. 345-380 | Article | MR 2205721 | Zbl 1088.22009

[32] Seven, Ahmet I. Recognizing cluster algebras of finite type, Electron. J. Combin., Tome 14 (2007) no. 1, pp. Research Paper 3, 35 pp. (electronic) | MR 2285804 | Zbl 1114.05103